Computing first-order and second-order determinants
Compute the following determinant.
$$\left|\begin{matrix} 1 & -2\\ 1 & -5 \end{matrix}\right|$$
$$\left|\begin{matrix} 1 & -2\\ 1 & -5 \end{matrix}\right|=1\cdot(-5)-(-2)\cdot1=-5+2=-3$$
$$-3$$
Invent a $2\times2$ matrix and compute its determinant.
$$A=\left(\begin{matrix} 1 & 4 \\ -2 & 3 \end{matrix} \right)$$
The determinant of the $2\times2$ matrix is the most immediate:
$$det(A)=\left|\begin{matrix} 1 & 4 \\ -2 & 3 \end{matrix} \right|=1\cdot3-4\cdot(-2)=3+8=11$$
$$det(A)=11$$